Let $K_v$ non-archim valued complete local field with finite residue field $\kappa= \mathcal{O}_K / \mathfrak{m}_v$ of characteristic $p$. Assume $K_v$ contains $d$th roots of unity $U_d$.
Is there sufficient numerical condition to decide when these roots of unity are canonically contained in the residue field $\kappa$?
The point is that even though because $\kappa^{\times} \cong \mathbb{F}_{p^f}^* $ for appropriate $f$ contains every cyclic group $C_d$ with $d \vert (p^f -1)$ by Lagrange, this not tells if the restricted residue map $\text{res}:\mathcal{O}_k^* \to \kappa^{\times}$ with kernel $1+ \mathfrak{m}_v$ embeds $U_d$ injectively into $\kappa$ making $U_d$ naturally $dth$ roots of unity in $\kappa$.
Is there any characterization when the res map restricted to $U_d$ is injective and therefore embeds the roots of unity into residue field, equivalently when $U_d \cap (1+ \mathfrak{m}_v)=\{1\}$?
You are going backwards: use Hensel's lemma to uniquely lift each nonzero element in $\kappa$ to a root of unity in $\mathcal O_K^\times$ that preserves its order.
There are no $p$-power roots of unity in a finite field, other than $1$, so the order of any root of unity in a finite field is not divisible by $p$. And they all lift to roots of unity in $\mathcal O_K^\times$.
Theorem. When $K$ is a local field with a residue field of size $q$, the polynomial $x^q - x$ splits completely in $\mathcal O_K[x]$ and its roots are distinct when reduces to the residue field.
Proof. Let $f(x) = x^q - x$ and $a \in \mathcal O_K$. Since all elements in $\kappa$ equal their own $q$th power and $q = 0$ in $\kappa$, we have $f(a) \equiv 0 \bmod \mathfrak m_K$ and $f'(a) = qa^{q-1} - 1 \equiv -1 \not\equiv 0 \bmod \mathfrak m_K$, where $\mathfrak m_K$ is the maximal ideal of $\mathcal O_K$. So by Hensel's lemma there is a unique root $\alpha$ of $f(x)$ in $\mathcal O_K$ such that $\alpha \equiv a \bmod \mathfrak m_K$. As $a$ runs over a set of representatives in $\mathcal O_K$ for $\kappa$, we obtain $q$ roots of $f(x)$ in $\mathcal O_K$ that are incongruent mod $\mathfrak m_K$, and $\deg f = q$, so we have obtained all the roots of $f(x)$. QED
Theorem. When $K$ is a local field with a residue field of characteristic $p$ and $m$ is a positive integer not divisible by $p$, two $m$th roots of unity in $\mathcal O_K$ that are equal in $\kappa$ must be equal in $\mathcal O_K$.
Proof. Let $z^m = 1$ in $\mathcal O_K$ and $f(x) = x^m - 1$, so $z \not\equiv 0 \bmod \mathfrak m_K$. Since $f(z) \equiv 0 \bmod \mathfrak m_K$ and $f'(z) = mz^{m-1} \not\equiv 0 \bmod \mathfrak m_K$, by Hensel's lemma there is a unique root $\alpha$ of $f(x)$ in $\mathfrak O_K$ such that $\alpha \equiv z \bmod \mathfrak m_K$. Since $z$ is such a root, $\alpha$ must be $z$: the only $m$th root of unity in $\mathcal O_K$ that's congruent to $z \bmod \mathfrak m_K$ is $z$. QED
If you understand the proofs of these two theorems, use similar ideas to show that when $\overline{a} \in \kappa^\times$ has order $d$ (so $d \mid (q-1)$), the unique $\omega$ in $\mathcal O_K$ that's a root of $x^q-x$ and satisfies $\overline{\omega} = \overline{a}$ in $\kappa$ must have order $d$.
Roots of unity in $K$ are in $\mathcal O_K^\times$, each nonzero coset in the residue field contains a unique $(q-1)$th root of unity by the first theorem above, and $p \nmid (q-1)$ (as $q$ is a power of $p$), so by the second theorem above the only roots of unity in $K$ with order relatively prime to $p$ are the $(q-1)$th roots of unity: $x^{q-1} - 1$ splits completely in $K[x]$ and its roots exhaust the prime-to-$p$ order roots of unity in $K$.
Example. When $p$ is prime, the prime-to-$p$ roots of unity in $\mathbf Q_p$ are the roots of $x^{p-1} - 1$, which has one root in each nonzero coset of the residue field, which has order $p$.
I have not said anything about nontrivial $p$-power roots of unity in a local field with residue field characteristic $p$. They could only occur in a characteristic $0$ local field (a finite extension of $\mathbf Q_p$) and they don't admit as straightforward a description as the prime-to-$p$ roots of unity, except they will all be in $1 + \mathfrak m_K$ since in the residue field the only $p$-power root of unity in a finite field with characteristic $p$. The $p$-power roots of unity in $\mathbf Q_p$ require turn out to be $\pm 1$ when $p = 2$ (note $-1 \equiv 1 \bmod 2$) and $\{1\}$ when $p > 2$.
Look up the Teichmuller lifting for more information about prime-to-$p$ roots of unity in local fields of characteristic $0$ with residue field characteristic $p$.